The national program to develop a trans-atmospheric vehicle has kindled a renewed interest in the modeling of supersonic reacting flows. A supersonic combustion ramjet, or scramjet, has been proposed to provide the propulsion system for this vehicle. The development of computational techniques for modeling supersonic reacting flow fields, and the application of these techniques to an increasingly difficult set of combustion problems are studied. Since the scramjet problem has been largely responsible for motivating this computational work, a brief history is given of hypersonic vehicles and their propulsion systems. A discussion is also given of some early modeling efforts applied to high speed reacting flows. Current activities to develop accurate and efficient algorithms and improved physical models for modeling supersonic combustion is then discussed. Some new problems where computer codes based on these algorithms and models are being applied are described.